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A small arithmetic hyperbolic three-manifold. (English) Zbl 0621.57006
It was shown by Jørgenson and Thurston that there is a minimal element \(v_ 1\) in the set of volumes of complete orientable hyperbolic three- manifolds. Let M be the complete orientable hyperbolic 3-manifold resulting from (5,1) Dehn surgery on the complement of the figure-eight knot K in \(S^ 3\). It is established in this paper that M is arithmetic. On the other hand, the author and Jørgenson have shown that there is an arithmetic manifold M’ with vol M’\(<vol M\).
Reviewer: G.Soifer

MSC:
57N10 Topology of general \(3\)-manifolds (MSC2010)
51M10 Hyperbolic and elliptic geometries (general) and generalizations
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
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